In light of the triviality of the fundamental group of varieties with the Hilbert property, the more refined “weak-Hilbert property” was introduced in [1]}, and it was shown in [2]} that abelian varieties with a dense set of \(K\) -points satisfy the weak-Hilbert property. Conjecturally, a variety with a dense set of \(K\) -rational points should have the weak-Hilbert property and even be special in the sense of Campana. In light of Campana's Abelianity Conjecture [3]}, we expect the topological fundamental group of a variety with a dense set of rational points to be virtually abelian (i.e., to contain a finite index abelian subgroup).
In particular, as smooth projective special varieties over \(\mathbb {C}\)  are expected to have a virtually abelian topological fundamental group, we also expect the topological fundamental group \(\pi _1(X_{\mathbb {C}})\)  of a smooth projective variety \(X\)  over a number field \(K\)  with a dense set of \(K\) -rational points to be virtually abelian.
Although this is far from known, there has been some progress on its function field analogue in [4]} building on recent progress on Lang's conjecture [5]}, [6]}.
